find number of automorphisms of (Z,+)

Question

What's the number of group isomorphisms from the group (Z, +) to itself??

my approach: I think it has to be infinity, but that is an incorrect answer...

Answer 1

Any homomorphism of a cyclic group is determined by what it does to a generator. So, by what it does to $1$.

But to be an automorphism, it has to be surjective. This won't happen unless $1$ is mapped to a generator.

So we need $\varphi(1)=1$, or $\varphi(1)=-1$. Thus there are two.